Question

A prism has total surface area of 360 m2 and volume of 60 m3.

Answer 1

Answer:

The answers are:

A. 7.5 m³

B. 90 m²

C. 180 m³

Step-by-step explanation:

Let us answer the questions about the volume. We know that the volume of a prism is V=l*w*h, where l stands for the length, w stands for the width and h stands for the height.

Volumes.

A prism with half the original sizes. From the original prism we know that V=l*w*h=60 m³. The new prism have dimension l'=l/2, h'=h/2 and w'=w/2. Then, its volume V' is

V' = l'*h'*w' = (l/2)*(h/2)*(w/2) = (l*w*h)/8=V/8 = 60/8 = 7.5 m³.

A prism with height tripled. In this case all the dimension are the same, except for the height: this l'=l, w'=w and h'=3*h. Then, the new volume V' is

V' = l'*h'*w' = l*(3*h)*w = 3*l*h*w=3*V = 3*60 = 180 m³.

Surface area.

A prism with half the original sizes.

The formula for the surface area of a prism is

$A = 2A_b + P_bh$

where $A_b$ stands for the area of the base and $P_b$ stands for the perimeter of the base. As the base is rectangle,  $A_b=lw$ and $P_b = 2(l+w)$.

Hence,

$A = 2lw + 2(l+w)h.$

Then, the area of the new prism is A' ()recall that the dimension of the new prism are l'=l/2, h'=h/2 and w'=w/2).

$A' = 2(l')(w') + 2(l'+w')h' = 2\frac{l}{2}\frac{w}{2} + 2(\frac{l}{2}+\frac{w}{2})\frac{h}{2} = \frac{2lw}{4} + 2\frac{l+w}{2}\frac{h}{2} = \frac{2lw}{4} + \frac{2(l+w)h}{4}.$

In this expression we can extract a common factor 1/4, thus

$A' = \frac{1}{4}(2lw + 2(l+w)h) = \frac{1}{4}A = 360/4 = 90 m^2$

Answer 2

1.
a. 180m²b. 7.5m³
2.a.1620m²b. 180m³